On the Calculation of Flywheels for Multicylinder Internal Combustion Engines (On the Question of Torsional Vibrations of Shafts)
Abstract Generated abstract
This paper addresses the selection of flywheel moment and shaft stiffness in multicylinder internal combustion engines when torsional vibration, rather than only rotational nonuniformity or stored kinetic energy, determines acceptable design. It develops a graphical and analytic procedure, based on the intersection of a straight line representing flywheel inertia and coupling stiffness with characteristic curves for the engine cylinder system, to determine natural torsional frequencies and shift dangerous critical speeds away from operating ranges. The article derives formulas for changing either flywheel inertia or the stiffness of the shaft section between engine and flywheel, discusses limits imposed by rotation uniformity and design constraints, and supplies data for eight-cylinder engines to extend the method.
Full Text
ON THE CALCULATION OF FLYWHEELS FOR MULTI-CYLINDER INTERNAL COMBUSTION ENGINES
(On the Problem of Torsional Vibrations of Shafts)
V. K. NECHAEV
Professor, Doctor of Technical Sciences
I
The required magnitude of the flywheel moment \(GD^2\) of the flywheel for a stationary engine is determined, as a rule, by the specified technical conditions for the degree of nonuniformity of shaft rotation \(\delta_s\). In individual cases, specific requirements are also taken into account with respect to fulfilling the conditions for stable parallel operation of a number of engines on a common alternating-current electrical network.
In automotive-tractor engines, the degree of nonuniformity \(\delta_s\), as is known, does not play a decisive role. Here the magnitude of \(GD^2\) of the flywheel is determined mainly by the need to accumulate a sufficient reserve of kinetic energy in the engine by the moment of clutch engagement, by the desire to ensure stable operation of the engine at small throttle openings and low speeds, etc.
But the frequencies of the shaft’s natural free torsional vibrations, and consequently all the critical numbers of revolutions corresponding to the sharp and strong development of resonant vibrations in the shaft, depend to a considerable extent on the magnitude of \(GD^2\) of the flywheel. Therefore, the calculation of the flywheel on the basis of the considerations indicated above cannot be considered final until the frequencies of the shaft’s natural vibrations have been calculated and the entire spectrum of critical numbers of revolutions has been determined from them.
Some of these critical numbers of revolutions may coincide with the engine’s operating numbers of revolutions, or may be inadmissibly close to the latter. Especially dangerous is the coincidence or close proximity of the engine’s operating speeds with critical speeds of low principal orders. In such cases there arises the problem of developing measures to combat strong vibrations of the engine shaft.
Eliminating strong resonant torsional vibrations of the shaft by appropriately shifting the engine’s operating speeds away from dangerous critical speeds is relatively rarely acceptable in stationary engines and, usually, has no real meaning at all in automotive-tractor engines, which operate over a very wide range of speeds.
During the design of a power plant, the simplest and most effective method of combating strong torsional vibrations of the shaft is to shift the engine’s critical revolutions. This shift is accomplished by a corresponding change in the shaft’s natural vibration frequencies through the selection of a new value of \(GD^2\) for the flywheel and a change in the stiffness of the shaft section between the last cylinder of the engine and the flywheel.
The necessary new values of the flywheel moment \(GD^2\), or of the moment of inertia of the flywheel mass \(\Theta_1\)¹) and of the shaft stiffness between the flywheel and the engine, are determined either by calculation, for example by the method of V. P. Terskikh²), or, more often, by tests. In both of these cases, however, the solution of the problem posed is connected with a large expenditure of time, owing to the need to investigate a whole series of possible variants.
The graphical method for determining the frequencies of the natural torsional vibrations of an engine shaft, described by the author in one of his previous papers,³) indicates ways of quickly and simply solving questions concerning the shift of critical speeds. This method makes it possible to obtain analytical expressions for those values of the moment of inertia \(\Theta_1\) of the flywheel and of the shaft stiffness between the engine and the flywheel which ensure a sufficient removal of the dangerous critical speeds from the specified operating speeds.
Such a problem is solved especially simply in installations with one large flywheel mass (Fig. 1). As a rule, we always have such a scheme in automotive-tractor engines and in installations of stationary engines operating on a slow-speed generator of the flywheel type, rigidly connected with the engine.
Fig. 1
In the author’s above-mentioned paper it was shown that the frequencies of the natural free vibrations of the engine—flywheel system (Fig. 1) are determined by the abscissas of the points of intersection of the curve \(D_m=f(A)\) with the straight line \(L=F(A)\) (Fig. 2). The equation of this straight line is
\[ L=\frac{A}{\beta_1}-\frac{1}{\alpha_1}, \tag{1} \]
where \(A\) is the principal parameter of the torsional vibrations of a multi-cylinder engine,
\[ A=\frac{\Theta\omega^2}{c}. \]
Here:
\[ \omega=\frac{\pi N}{30}\ \mathrm{sec}^{-1}; \]
\(\omega\) is the angular velocity of the vibrations (angular frequency), connected with the vibration frequency per minute \(N\) by the relation
\(\Theta\) is the moment of inertia of the reduced flywheel masses of each engine cylinder;
\(c\) is the stiffness of the section of the crankshaft between the axes of each two neighboring cylinders, measured, for example, in \(\mathrm{kg\,cm/radian}\),⁴)
\[ \alpha_1=\frac{\Theta_1}{\Theta}, \]
¹) The quantities \(GD^2\) and \(\Theta_1\) are connected with one another by the generally known relation:
\[ GD^2=4g\cdot\Theta_1, \]
where \(g\) is the acceleration of gravity.
²) V. P. Terskikh, Torsional Vibrations of Power Installations, Book 1, 1940.
³) V. K. Nechaev, “Graphical Method for Determining Frequencies of Natural Vibrations,” Diesel Engineering, 1934, No. 4, and also Proceedings of the Tomsk Industrial Institute, vol. 54, issue 6.
⁴) Here, as everywhere below, it is assumed that the reduced lengths of the crankshaft sections between the axes of each two neighboring cylinders, and consequently also the stiffnesses \(c\) of these sections, are equal to one another. This condition is fulfilled in all sta-
$\Theta_1$—the moment of inertia of the mass of the flywheel,
\[ \beta_1=\frac{c_{m1}}{c}, \]
$c_{m1}$—the stiffness of the shaft section between the last cylinder of the engine and the flywheel.
The coefficient $D_m$, characterizing the ratio of the amplitudes of the free oscillations of the reduced masses of the individual cylinders of the engine, is an algebraic function of $A$, determined entirely by the number of cylinders in the engine (for identical $\Theta$ and $c$ for all cylinders). The coefficient $D_m$ is completely independent of the specific dimensions of the machine. Thus, for four-cylinder engines
\[ D_4=\frac{1-6A+5A^2-A^3}{4-10A+6A^2-A^3}, \tag{2} \]
and for six-cylinder engines
\[ D_6=\frac{1-15A+35A^2-17A^3+8A^4-A^5}{6-35A+56A^2-25A^3+9A^4-A^5}. \tag{3} \]
Graphically, the dependence $D_m=f(A)$ in the coordinates $D_m$ and $A$ is represented by a discontinuous curve consisting of a number of branches1.
The straight line $L$, determined by equation (1), cuts off on the ordinate axis (Fig. 2) the segment $-\dfrac{1}{\alpha_1}$, numerically equal to the ratio of the moments of inertia $\Theta$ and $\Theta_1$. The tangent of the angle of inclination of this straight line to the abscissa axis is $1/\beta_1$, i.e., it depends only on the ratio of the stiffnesses $c$ and $c_{m1}$.
When determining the natural frequencies of the system according to Fig. 1, the quantities $\Theta$, $\Theta_1$, $c$, $c_{m1}$, $a$, and consequently also the coefficients $\alpha_1$ and $\beta_1$, which characterize the position of the straight line $L$ in Fig. 2, are known in advance.
The abscissa $A_1$ of the point $M_1$ of intersection of the straight line $L$ with the first branch of the curve $D_m$ gives the frequency of the single-node free oscillations of the system
\[ N_1=\frac{30}{\pi}\sqrt{\frac{A_1\cdot c}{\Theta}}\ \text{oscillations/min}, \tag{4} \]
which, as a rule, is of primary interest to the designer when detecting the presence or absence of dangerous torsional oscillations in the working speed range of the installation being designed.
The abscissa $A_2$ of the point $M_2$ of intersection of the straight line $L$ with the second branch of the curve $D_m$ gives the frequency of the two-node free oscillations of the shaft:
\[ N_{II}=\frac{30}{\pi}\sqrt{\frac{A_2\cdot c}{\Theta}}\ \text{oscillations/min}. \tag{5} \]
stationary and many lightweight engines. With unequal reduced lengths of the shaft between adjacent cylinders, one may, without great error—at least in the calculation of single-node oscillations—understand by $c$ the mean value determined by the formula
\[ \frac{1}{c}=\frac{1}{m-1}\left(\frac{1}{c_{12}}+\frac{1}{c_{23}}+\ldots+\frac{1}{c_{m-1,m}}\right). \]
Here $m$ is the number of cylinders of the engine, and, for example, $c_{23}$ is the actual stiffness of the section of the crankshaft between the axes of the second and third cylinders.
Varying the quantities \(\Theta_1\) and \(c_{m1}\), we shall thereby change the position of the straight line \(L\) on the diagram (Fig. 2), obtaining different points of its intersection with the curve \(D_m\), and consequently also different natural frequencies of the system in Fig. 1. These properties of the diagram in Fig. 2 also make it possible to solve the inverse problem of finding a new straight line \(L'\), intersecting the curve \(D_m\) at points specified by the designer, i.e., ensuring the obtaining of such concrete values of the natural frequencies for which
Fig. 2
there will be achieved a sufficient removal of the most dangerous critical speeds from the operating speeds of rotation of the engine shaft.
Suppose that, in order to shift the critical speeds (primarily those of the principal orders), it is necessary to have a new frequency of natural single-node oscillations \(N_1'\) instead of the previously calculated \(N_1\). The frequency \(N_1'\) corresponds to the value of the parameter \(A\):
\[ A_1'=\frac{\Theta \omega_1'^2}{c}, \tag{6} \]
where
\[ \omega_1'=\frac{\pi N_1'}{30}. \]
The magnitude \(A_1'\) gives the abscissa of that point \(M_1'\) on the first branch of the curve \(D_m\) (Fig. 2) through which the straight line \(L'\) must pass. The ordinate \(D_{m1}\) of the point \(M_1'\) may be calculated, for example, from formulas (2) or (3), or taken directly from the table.
But the equation of the straight line \(L'\) is still not determined by the coordinates of only its single point \(M_1'\). This indicates the obvious circumstance that the problem of changing the frequency of the natural single-node oscillations of the system under consideration, with the aim of shifting the critical rotational speeds, has many solutions, corresponding to the many possible combinations of the quantities \(\Theta_1\) and \(c_{m1}\).
Let us consider particular cases.
- Shift of the critical rotational speeds by changing \(\Theta_1\) with unchanged stiffness \(c_{m1}\).
In this case the new straight line \(L'\) will pass through the point \(M_1'\), being parallel to the straight line \(L\) constructed earlier for the initially adopted characteristics \(\alpha_1\) and \(\beta_1\) of the system.
Draw on the diagram (Fig. 2) the straight line \(A_1'b\), parallel to the straight lines \(L\) and \(L'\). Then from the drawing:
\[ \overline{Ob}=A_1'\operatorname{tg}\Psi=\frac{A_1'}{\beta_1}=\frac{1}{\alpha_1'}+D'_{m_1}, \]
where
\[ \overline{OB_1}=\frac{1}{\alpha_1'} \]
is the segment cut off by the straight line \(L'\) on the ordinate axis, while \(\beta_1\) is a known coefficient, calculated earlier, which remains unchanged in the present variant in accordance with the preservation of the rigidity \(c_{m1}\). Hence
\[ \overline{OB_1}=\frac{1}{\alpha_1'}=\frac{A_1'}{\beta_1}-D'_{m_1}. \]
Now the new value of the moment of inertia \(\Theta_1'\) of the flywheel, for which the installation according to Fig. 1 will have the natural frequency of single-node vibrations equal to \(N_1'\), is
\[ \Theta_1'=\alpha_1'\cdot\Theta=\Theta\cdot\frac{\beta_1}{A_1'-\beta_1D'_{m_1}}, \tag{7} \]
where, as before, \(\Theta\) is the moment of inertia of the reduced flywheel masses of each cylinder.
Analysis of the diagram (Fig. 2) indicates the comparative limitation of the possibilities available to the designer, by changing the moment of inertia \(\Theta_1\) of the flywheel, for changing the natural frequencies of single-node vibrations of the installation shaft. It is obvious that, theoretically, by varying \(\Theta_1\) within the limits from 0 to \(\infty\), one can obtain the following interval of natural frequencies of single-node vibrations:
\[ \text{from}\quad N_{1\min}=\frac{30}{\pi}\sqrt{\frac{A_{1\min}\cdot c}{\Theta}} \]
\[ \text{to}\quad N_{1\max}=\frac{30}{\pi}\sqrt{\frac{A_{1\max}\cdot c}{\Theta}}. \]
Here \(A_{1\min}\) is the abscissa of the point of intersection of the straight line \(L''\), drawn from the origin (Fig. 2) parallel to the straight line \(L\), with the curve \(D_m\), corresponding to the case:
\[ \frac{1}{\alpha_1'}=0\quad \text{or}\quad \Theta_1=\infty, \]
and \(A_{1\max}\) is the abscissa of the point of discontinuity between the first and second branches of the curve \(D_m\).
As a result of calculating \(\Theta_1'\) by formula (7) for various values of \(N\) within the limits from \(N_{1\min}\) to \(N_{1\max}\), one can construct the diagram shown in Fig. 3. On the same diagram it is also advisable to plot the corresponding values of the degree of irregularity \(\delta_s\) of an absolutely rigid shaft, calculated, for example, by the usual formula
\[ \delta_s=\frac{3600A_{\text{exc}}}{n^2\cdot 4g\,(\Theta_1'+m\Theta)}, \tag{8} \]
where \(n\) is the normal number of revolutions of the engine per minute, and \(A_{\text{exc}}\) is the excess work, determined from the diagram of tangential forces on the engine shaft.
Of course, this curve \(\delta_s=\varphi(N_1)\) has only a very conditional significance, since, owing to torsional vibrations, the actual degree of nonuniformity \(\delta\) of the working end of the shaft may differ considerably from \(\delta_s\) ^1). But this difference is especially substantial only in the presence of strong torsional vibrations arising in speed ranges close to strong critical speeds, i.e., precisely in those cases whose elimination is the purpose of the calculations set forth in this article.
The diagram in Fig. 3 makes it possible, without great difficulty, to choose such a value of the moment of inertia \(\Theta_1\) of the flywheel as will ensure the required shift of the critical speeds while maintaining a satisfactory \(\delta_s\). By marking on the \(\delta_s\) curve the maximum value of \(\delta_s\) still permissible under the technical conditions for the given installation, we thereby single out the region of those values of \(\Theta_1\), and consequently also of the frequencies of the natural one-node vibrations \(N_1\), which should be considered unacceptable from the standpoint of ensuring sufficient uniformity of rotation of the machine shaft. In automotive-tractor installations, the limits of the possible variants for choosing \(\Theta_1\) of the flywheel under the conditions of the necessary shift of the critical speeds, while maintaining a sufficient minimum of kinetic energy, are usually narrowed still more appreciably by design and dimensional considerations.
Fig. 3
In these cases it is expedient to investigate the possibility of changing \(N_1\) by changing the stiffness \(C_{m1}\) of the shaft between the engine and the flywheel while preserving the previously selected value \(\Theta_1\) of the flywheel.
2. Shift of critical rotational speeds by changing the stiffness \(C_{m1}\) at \(\Theta_1=\mathrm{const}\).
When \(C_{m1}\) is changed, the straight line \(L\) rotates about the point \(B\) (Fig. 4) in accordance with the change in the ratio
\[ \frac{1}{\beta}=\frac{C}{C_{m1}}, \]
which represents the tangent of the angle of inclination of this straight line to the axis of abscissae. The new straight line \(L'\) that we need, giving the frequency \(N_1'\), must pass through the points \(B\) and \(M_1'\).
We draw the auxiliary straight line \(A_1'b\), parallel to the straight line \(L'\). Then, from the drawing:
\[ \operatorname{tg}\Psi'=\frac{1}{\beta_1'}= \frac{\overline{OB}+D'_{m_1}}{A_1'}= \frac{\frac{1}{\alpha_1}+D'_{m_1}}{A_1'} . \]
Hence we obtain an expression for calculating the new required value \(C'_{m1}\) of the stiffness of the shaft between the last cylinder of the engine and the flywheel,
^1) The dependence of the actual degree of nonuniformity \(\delta\) on the frequency of natural vibrations \(N_1\), with a constant engine speed \(n\), is of considerably more complex character. Here it is already necessary to introduce the concept of the critical value of the flywheel moment of inertia and to solve the problem of choosing the flywheel by other methods. (M. V. Nechaev—“Theoretical Torsiograms for the Shaft of an Internal Combustion Engine,” Proceedings of the Tomsk Industrial Institute named after S. M. Kirov, vol. 58, issue II.)
at which the arrangement in Fig. 1 has the natural single-node vibration frequency \(N_1'\):
\[ c'_{m1}=\beta'_1\cdot c=c\cdot \frac{A'_1}{\frac{1}{a_1}+D'_{m1}}, \tag{9} \]
where, as before, \(D'_{m1}\) is the ordinate of the point \(M'_1\), having the abscissa \(A'_1\), calculated by formula (6).
Fig. 4 shows that, by varying \(c_{m1}\), one can obtain, at least theoretically, a wide range of values of \(N'_1\) within the limits from
\[ N_{1min}=\frac{30}{\pi}\sqrt{\frac{A_{1min}\cdot c}{\Theta}} \]
to
\[ N_{1max}=\frac{30}{\pi}\sqrt{\frac{A_{1max}\cdot c}{\Theta}}, \]
where \(A_{1max}\) is determined as the abscissa of the point \(M''_1\) of intersection of the curve \(D_m\) with the straight line \(L''\), drawn from the point \(B\) at the minimum possible angle \(\psi_{min}=\operatorname{arctg} 1/\beta_{max}\). The value \(\beta_{max}\) may be chosen on the basis of
Fig. 4
the minimum structurally acceptable length of the shaft between the axis of the last cylinder and the flywheel, with its maximum structurally convenient diameter. The value \(A_{1min}\), corresponding to the limiting theoretical case \(c_{m1}=0\), is obviously equal to zero.
3. Shift of the critical speeds by simultaneous variation of \(C_{m1}\) and \(\Theta_1\).
When it is necessary to obtain a considerable shift of the critical speeds—for example, in automotive-tractor engines—it is sometimes most expedient to use simultaneously the available possibilities of changing the moment of inertia \(\Theta_1\) of the flywheel and the stiffness \(C_{m1}\) of the shaft between the engine and the flywheel. Now, in the diagram (Fig. 5), both the initial point and the angle of inclination of the straight line \(L\) to the abscissa axis will change.
Assigning, for example, a new, structurally acceptable value \(\Theta'_1\), which also ensures a sufficient minimum of the kinetic energy of the engine or the required \(\delta_s\), one can, by formula (9), find the corresponding
the corresponding value, \(C_{mI}'\), at which the installation being calculated will have the prescribed frequency of natural single-node vibrations \(N_I'\). Similarly, by formula (7) the inverse problem is also solved: determining the necessary value of \(\Theta_I'\) for a new value of \(C_{mI}'\), adopted in accordance with design considerations.
Thus, formulas (7) and (9) make it possible quickly to determine those new values \(\Theta_I\) and \(C_{mI}\) which, of course within the limits of the possibilities considered above, will ensure the required value \(N_I'\). This new required frequency of single-node vibrations \(N_I'\) is chosen with such a calculation that none of the dangerous critical speeds coincides with the operating speeds of the engine and is not located close to the latter.
However, ensuring the desired \(N_I'\) by the appropriate choice of \(\Theta_I\) and \(C_{mI}\) still does not completely eliminate all dangers of torsional resonances arising in the range of the engine operating revolutions. In individual cases, resonances corresponding to two-node vibrations of the shaft with frequency \(N_{II}\) may also prove dangerous. This frequency is determined by the abscissa \(A_2\) of the point of intersection of the straight line \(L\) with the second branch of the curve \(D_m\) (formula 5).
Fig. 5
The method of calculation set forth above indicates ways of determining those values \(\Theta_I\) and \(C_{mI}\), in the installation according to Fig. 1, for which the engine operating-speed range will be free from dangerous resonant vibrations, both single-node and two-node.
Let us suppose that, for such a clearing of the region of the engine operating revolutions, it is necessary that the installation under consideration according to Fig. 1 have frequencies \(N_I'\) and \(N_{II}'\), respectively, of single-node and two-node free vibrations. We calculate the values of the parameter
\[ A_1'=\frac{\Theta}{c}\left(\frac{\pi N_I'}{30}\right)^2, \]
\[ A_2'=\frac{\Theta}{c}\left(\frac{\pi N_{II}'}{30}\right)^2 \]
and from the points \(A_1'\) and \(A_2'\) we raise ordinates to their intersections with the corresponding branches of the curve \(D_m\) at the points \(M_1'\) and \(M_2'\) (Fig. 5). Through these points the new straight line \(L'\) must pass. From the drawing:
\[ \tg\psi'=\frac{1}{\beta_1'}=\frac{D_{m_2}'-D_{m_1}'}{A_2'-A_1'}, \]
where \(D_{m_1}'\) and \(D_{m_2}'\) are the ordinates of the points \(M_1'\) and \(M_2'\). The numerical values of these ordinates can be calculated from equations (2), (3) or taken from tables.
Hence:
\[ \beta_1'=\frac{A_2'-A_1'}{D_{m_2}'-D_{m_1}'} \]
and
\[ c_{m\mathrm{I}}'=\beta_1' c = c\,\frac{A_2'-A_1'}{D_{m_2}'-D_{m_1}'} . \tag{10} \]
Moreover, according to Fig. 5,
\[ \frac{1}{\alpha_1'}+D_{m_1}' = A_1 \tan \Psi' = \frac{1}{\beta_1'} A_1' = A_1'\,\frac{D_{m_2}'-D_{m_1}'}{A_2'-A_1'}, \]
or
\[ \frac{1}{\alpha_1'} = \frac{A_1'(D_{m_2}'-D_{m_1}')}{A_2'-A_1'} - D_{m_1}' . \]
Now
\[ \Theta_1'=\alpha_1'\Theta = \frac{A_2'-A_1'}{A_1'D_{m_2}'-A_2'D_{m_1}'}\,\Theta . \tag{11} \]
Formulas (10) and (11) give those specific values of \(\Theta_1\) and \(c_{m\mathrm{I}}\) for which the installation according to Fig. 1 will have the prescribed natural frequencies \(N_{\mathrm{I}}'\) and \(N_{\mathrm{II}}'\).
II
Above we considered a method for calculating the shift of the critical speeds in the installation according to Fig. 1, which has one flywheel. But the problem of the shift can be solved in an entirely analogous manner also in installations with two flywheels (for example, engine—flywheel—dynamo, Fig. 6). Here, however, the dependence \(L=F(A)\) proves to be more complicated, and in the coordinates \(D_m-A\), instead of a straight line, a discontinuous curve is obtained.1
Fig. 6
To change the natural frequencies of the shaft of such an installation, one may vary (separately or simultaneously) the stiffness \(c_{m\mathrm{I}}\) of the shaft between the engine and the flywheel, the stiffness \(c_{\mathrm{I}\mathrm{II}}\) of the shaft between the flywheel and the dynamo, and the moments of inertia \(\Theta_{\mathrm{I}}\) and \(\Theta_{\mathrm{II}}\), respectively, of the flywheel and the dynamo rotor.
Thus, in order to obtain a prescribed frequency of free single-node vibrations \(N_{\mathrm{I}}'\), with the accepted \(\Theta_{\mathrm{I}}, \Theta_{\mathrm{II}}\) and \(c_{\mathrm{I}\mathrm{II}}\), it is necessary to have the following stiffness of the shaft between the last cylinder of the engine and the flywheel:2
\[ c_{m\mathrm{I}}' = c\, \frac{A_1'} { D_{m_1}' - \frac{\beta_2-\alpha_2 A_1'} {\alpha_1\alpha_2 A_1'-(\alpha_1+\alpha_2)\beta_2} }, \tag{12} \]
where, as before,
\[ \alpha_1=\frac{\Theta_1}{\Theta}. \]
and, moreover:
\[ \alpha_2=\frac{\Theta_{\mathrm{II}}}{\Theta}, \qquad \beta_2=\frac{c_{\mathrm{II}}}{c}. \]
The quantity \(D_{m_1}'\) entering here is the ordinate of that point \(M_1'\) on the curve \(D_m=f(A)\) through which the curve \(L=F(A)\) must pass. The abscissa \(A_1'\) of this point is determined by equation (5).
The system according to Fig. 6 will have a frequency of its natural single-node oscillations equal to \(N_1'\) if, for given \(\Theta_{\mathrm{I}}, \Theta_{\mathrm{II}}, c_{m\mathrm{I}}\), the stiffness of the connecting shaft between the flywheel and the dynamo rotor is made of the following magnitude:
\[ c_{\mathrm{I,II}}'=\alpha_2 c A_1'\cdot \frac{\alpha_1(A_1'-\beta_1D_{m_1}')-\beta_1} {(\alpha_1+\alpha_2)(A_1'-\beta_1D_{m_1}')-\beta_1}. \tag{13} \]
The installation according to Fig. 6 will have frequency \(N_1'\) for given \(c_{m\mathrm{I}}, c_{\mathrm{I,II}}\), and \(\Theta_{\mathrm{II}}\), if a flywheel is installed with moment of inertia
\[ \Theta_{\mathrm{I}}'=\Theta\cdot \left( \frac{\alpha_2\beta_2}{A_1'\alpha_2-\beta_2} + \frac{\beta_1}{A_1'-\beta_1D_{m_1}'} \right). \tag{14} \]
Finally, the specified frequency of the single-node natural oscillations \(N_1'\) of the installation according to Fig. 6 can also be obtained by choosing the corresponding value of the moment of inertia \(\Theta_{\mathrm{II}}\) of the dynamo rotor. Here, for the adopted \(\Theta_{\mathrm{I}}, c_{m\mathrm{I}}, c_{\mathrm{I,II}}\), it is necessary to have
\[ \Theta_{\mathrm{II}}'=\beta_2\Theta\cdot \frac{ 1+\alpha_1\left(D_{m_1}'-\dfrac{A_1'}{\beta_1}\right) }{ A_1'+(\alpha_1A_1'-\beta_2)\left(D_{m_1}'-\dfrac{A_1'}{\beta_1}\right) }. \tag{15} \]
Of course, each of these formulas (12)—(15) can in practice be used only so long as the values of the moment of inertia or stiffness determined by them are quantities that can be implemented easily and conveniently in design. Otherwise, the prescribed shift of the critical rotational speeds must be achieved by simultaneously changing at least two characteristics of the system, for example, \(c_{m\mathrm{I}}\) and \(c_{\mathrm{I,II}}\), or \(c_{m\mathrm{I}}\) and \(\Theta_{\mathrm{I}}\), etc.
III
The first branch of the curve \(D_m=f(A)\), determined for four- and six-cylinder engines by equations (2) and (3), can be represented by the following approximate formula:
\[ D_m=e-\frac{f}{g-A}, \tag{16} \]
where \(e, f\), and \(g\) are constants depending only on the number of cylinders of the engine.
Thus, for 4-cylinder engines, in the interval from \(A=0\) to \(A=0.30\):
\[ e=0.7877;\qquad f=0.335;\qquad g=0.623. \]
For 6-cylinder engines, in the interval from \(A=0\) to \(A=0.12\):
\[ e=0.620;\qquad f=0.136;\qquad g=0.300. \]
With these values of the coefficients \(e, f, g\) and in the indicated intervals, formula (15) gives values of \(D\) accurate to one unit in the third decimal place.
Now the abscissa \(A_1\) of the point \(M_1\) of intersection of the curve \(D_m\) with the straight line \(L\) (Fig. 2) is determined by the simultaneous solution of equation (16) with equation (1) for \(L\):
\[ \frac{A}{\beta_1}-\frac{1}{\alpha_1}=e-\frac{f}{g-A}. \]
Hence, after elementary transformations,
\[ A^2-2Ap+q=0, \tag{17} \]
where
\[ p=\frac{g+\dfrac{\beta_1}{\alpha_1}+\beta_1 e}{2}; \]
\[ q=g\frac{\beta_1}{\alpha_1}+eg\beta_1-\beta_1 f. \]
Substituting here the values of the constants \(e, f, g\) given above, we obtain:
for 4-cylinder engines
\[ p=\frac{\beta_1}{2\alpha_1}+0.394\beta_1+0.311; \]
\[ q=0.623\frac{\beta_1}{\alpha_1}+0.156\beta_1 \tag{18} \]
and for 6-cylinder engines:
\[ p=\frac{\beta_1}{2\alpha}+0.310\beta+0.150; \]
\[ q=0.300\frac{\beta_1}{\alpha_1}+0.05\beta. \tag{19} \]
The smaller root of equation (17) gives the required abscissa of the point \(M_1\) (Fig. 2):
\[ A_1=p-\sqrt{p^2-q}. \tag{20} \]
Hence, on the basis of formula (4), we have an expression for the frequency of the natural (free) single-node vibrations of the installation according to Fig. 1:
\[ N_{\mathrm{I}}=\frac{30}{\pi}\sqrt{\frac{c}{\Theta}}\cdot \sqrt{p-\sqrt{p^2-q}}. \tag{21} \]
The error in calculating \(N_{\mathrm{I}}\) by this formula does not exceed \(0.25\%\). Thus, formulas (18)—(21) provide an accuracy of computation of \(N_{\mathrm{I}}\) that is entirely sufficient for all calculations performed on a slide rule. The use of these formulas presents no difficulties and is not associated with carrying out complex computations.
By calculating \(N_{\mathrm{I}}\) from formulas (20)—(21), one can very quickly check the suitability, for example, of the adopted value of the moment of inertia \(\Theta_1\) of the flywheel from the standpoint of avoiding coincidence of any critical speeds corresponding to single-node vibrations with the operating speeds of rotation of the engine shaft.
The second, larger root of equation (17) gives the abscissa of the point of intersection of the straight line \(L\) (Fig. 2) with the second branch of the curve \(D_m\), determining the frequency \(N_{\mathrm{II}}\) of the natural two-node vibrations of the shaft of the installation according to Fig. 1. However, the accuracy of such a calculation of \(N_{\mathrm{II}}\) will be low, since the second
branch of the curve \(D_m\) fits formula (17) poorly. The frequency \(N\) must be calculated graphically, by determining the abscissa \(A_2\) of the point \(M\) by direct construction of the straight line on the diagram \(D_m=f(A)\).
Table 1
Relative amplitudes of oscillations of the 7th and 8th masses of an eight-cylinder engine
\[ \varphi_1=+1,0000\ \text{rad.} \qquad A=\frac{\Theta\omega^2}{c} \]
| \(A\) | \(\varphi_7\) | \(\varphi_8\) | \(A\) | \(\varphi_7\) | \(\varphi_8\) | \(A\) | \(\varphi_7\) | \(\varphi_8\) |
|---|---|---|---|---|---|---|---|---|
| 0 | +1,0000 | +1,0000 | 0,16 | −0,8837 | −1,0131 | 0,32 | −0,8685 | −0,4166 |
| 0,01 | +0,7969 | +0,7324 | 0,17 | −0,9226 | −1,0216 | 0,33 | −0,8338 | −0,3505 |
| 0,02 | +0,6073 | +0,4837 | 0,18 | −0,9567 | −1,0211 | 0,34 | −0,7996 | −0,2836 |
| 0,03 | +0,4308 | +0,2679 | 0,19 | −0,9832 | −1,0125 | 0,35 | −0,7572 | −0,2161 |
| 0,04 | +0,2738 | +0,0686 | 0,20 | −1,0035 | −0,9963 | 0,36 | −0,7156 | −0,1484 |
| 0,05 | +0,1148 | −0,1102 | 0,21 | −1,0178 | −0,9732 | 0,37 | −0,6722 | −0,0806 |
| 0,06 | −0,0256 | −0,269 | 0,22 | −1,0266 | −0,9437 | 0,38 | −0,6271 | −0,0132 |
| 0,07 | −0,1517 | −0,4108 | 0,23 | −1,0300 | −0,9085 | 0,39 | −0,5805 | +0,0537 |
| 0,08 | −0,2732 | −0,5346 | 0,24 | −1,0285 | −0,8681 | 0,40 | −0,5325 | +0,1198 |
| 0,09 | −0,3813 | −0,6420 | 0,25 | −1,0222 | −0,8231 | 0,41 | −0,4835 | +0,1848 |
| 0,10 | −0,4796 | −0,7341 | 0,26 | −1,0115 | −0,7738 | 0,42 | −0,4334 | +0,2487 |
| 0,11 | −0,5684 | −0,8118 | 0,27 | −0,9966 | −0,7209 | 0,43 | −0,3825 | +0,3112 |
| 0,12 | −0,6481 | −0,8759 | 0,28 | −0,9781 | −0,6648 | 0,44 | −0,3310 | +0,3720 |
| 0,13 | −0,7191 | −0,9272 | 0,29 | −0,9554 | −0,6059 | 0,45 | −0,2789 | +0,4311 |
| 0,14 | −0,7818 | −0,9667 | 0,30 | −0,9295 | −0,5446 | |||
| 0,15 | −0,8365 | −0,9951 | 0,31 | −0,9005 | −0,4814 |
Note: The values of \(\varphi\) for the first six masses of the engine—see the author’s work, “Theoretical torsiograms for the shaft of an internal-combustion engine,” Izvestia TII, vol. 58 (1937).
Table 2
Values of \(\displaystyle \sum_1^8 \varphi_i\) and \(D_8\) for an eight-cylinder engine
\[ \varphi_1=+1,0000\ \text{rad.}; \qquad D_8=\frac{\varphi_8}{\displaystyle \sum_1^8 \varphi_i} \]
| \(A\) | \(\displaystyle \sum_1^8 \varphi_i\) | \(D_8\) | \(A\) | \(\displaystyle \sum_1^8 \varphi_i\) | \(D_8\) | \(A\) | \(\displaystyle \sum_1^8 \varphi_i\) | \(D_8\) |
|---|---|---|---|---|---|---|---|---|
| 0 | +8,0000 | +0,1215 | 0,16 | −0,2042 | +4,960 | 0,32 | −1,8288 | +0,2278 |
| 0,01 | +7,1848 | +0,1019 | 0,17 | −0,4455 | +2,2930 | 0,33 | −1,8150 | +0,1931 |
| 0,02 | +6,4182 | +0,076 | 0,18 | −0,6631 | +1,5400 | 0,34 | −1,7925 | +0,1582 |
| 0,03 | +5,6936 | +0,0470 | 0,19 | −0,8582 | +1,1800 | 0,35 | −1,7620 | +0,1226 |
| 0,04 | +5,0296 | +0,0136 | 0,20 | −1,0317 | +0,9557 | 0,36 | −1,7242 | +0,0860 |
| 0,05 | +4,3901 | −0,0251 | 0,21 | −1,1857 | +0,8207 | 0,37 | −1,6794 | +0,0180 |
| 0,06 | +3,7967 | −0,0710 | 0,22 | −1,3203 | +0,7158 | 0,38 | −1,6287 | +0,0081 |
| 0,07 | +3,2468 | −0,1265 | 0,23 | −1,4368 | +0,6323 | 0,39 | −1,5724 | −0,0341 |
| 0,08 | +2,7326 | −0,1956 | 0,24 | −1,5363 | +0,5651 | 0,40 | −1,5110 | −0,0793 |
| 0,09 | +2,2546 | −0,2848 | 0,25 | −1,6197 | +0,5082 | 0,41 | −1,4452 | −0,1279 |
| 0,10 | +1,8112 | −0,405 | 0,26 | −1,6880 | +0,4584 | 0,42 | −1,3754 | −0,1808 |
| 0,11 | +1,4009 | −0,5794 | 0,27 | −1,7420 | +0,4138 | 0,43 | −1,3020 | −0,2390 |
| 0,12 | +1,0229 | −0,8563 | 0,28 | −1,7827 | +0,3729 | 0,44 | −1,2257 | −0,3035 |
| 0,13 | +0,6738 | −1,3761 | 0,29 | −1,8110 | +0,3346 | 0,45 | −1,1467 | −0,3760 |
| 0,14 | +0,3511 | −2,7300 | 0,30 | −1,8276 | +0,2980 | |||
| 0,15 | +0,0619 | −16,0860 | 0,31 | −1,8332 | +0,2626 |
OBSERVED MISPRINTS
| Page | Line | Printed | Should be |
|---|---|---|---|
| 7 | 3 from bottom | result | results |
| 9 | 2 from top | \(\alpha = h\eta_0 t\) | \(\alpha = \eta_0 t\) |
| 10 | 1 from bottom | \(\delta_s \cong b_i\) | \(\delta_s \cong \delta_i\) |
| 19 | 6 from top | \(+\displaystyle\sum_{q=0}^{\infty}(-1)^q J_{2q}(\xi)+\) | \(+\displaystyle\sum_{q=0}^{\infty}(-1)^q J_{2q}(\xi)+\) |
| 19 | 7 from top | \(-J2q+3(\xi)]\) | \(-J_{2q+3}(\xi)]\) |
| 21 | 4 from bottom | \([J_0(\xi_1)\sin(\eta_h t+\) | \([J_0(\xi_1)\sin \eta_h t+\) |
| 22 | 7 from top | \(\sin(\eta_h-2\nu)t\) | \(\sin(\eta_h-2\nu)t+\ldots\) |
| 25 | 3 from bottom | \(\left[1+\dfrac{\delta_s}{2}\cos(\nu t-\zeta)\right]\) | \(\left[1+\dfrac{\delta_s}{2}\cos(\nu t-\zeta)\right]^2\) |
| 27 | 3 from top | \(C=-\dfrac{\delta_s^2}{16}\{\) | \(C=\dfrac{\delta_s^2}{16}\{\) |
| 30 | 25 from top | varying | decreasing |
| 31 | 2 from top | (69) | (65) |
| 31 | 13 from bottom | \(m_9\) | \(m_3\) |
| 34 | 12 from top | \(\displaystyle\sum_{p=1}^{\infty} M_p\sin(p\alpha-\varepsilon_p')\) | \(\displaystyle\sum_{p=1}^{\infty} M_p\sin(p\alpha+\varepsilon_p')\) |
| 38 | 3 from bottom | terms | term |
| 42 | 20 from top | \(k\ k_c\) | \(k/k_c\) |
| 47 | 14 from top | \(\Psi=\Phi_{q-1}\) | \(\Psi=\displaystyle\sum_{q=1}^{\infty}\Phi_{q-1}\) |
| 48 | 12 from top | \(M_h J_1\!\left(\dfrac{\delta}{2}\right)\) | — |
| 51 | 5 from top | \(\dfrac{\delta}{2}\sin\eta_k t\) | \(\dfrac{\delta}{2}\sin\eta_h t\) |
| 52 | 23 from top | \(\Phi_0\) | \(\Phi_{0res}\) |
| 54 | 5 from top | accept | recognize |
| 55 | 17 from top | \(\displaystyle\sum_{q=1}^{\infty}(-1)d\) | \(\displaystyle\sum_{q=1}^{\infty}(-1)^q\) |
| 57 | 2 from bottom | \(M_{kp}'\) | \(M_{kp}''\) |
| 58 | 3 from bottom | \(\overline{B}_h=\overline{B}_h'=\overline{B}_h''\) | \(\overline{B}_h=\overline{B}_h'+\overline{B}_h''\) |
| 64 | 4 from top | \(\overline{M}_h\sin(h\eta_0t-\overline{\varepsilon}_h)\) | \(\overline{M}_h\sin(h\eta_0t+\overline{\varepsilon}_h)\) |
| 79 | 4 from bottom | \(+1{,}0229\) | \(+1{,}0223\) |
| 87 | 2 from bottom | \((\varphi_{i\,rel}-\varphi_{i+1,\,rel}^{2})\) | \((\varphi_{i\,rel}-\varphi_{i+1,\,rel})^{2}\) |
| 90 | 6 from top | \(M_h'=\) | \(M_r'=\) |
| 92 | 5 from top | \(+c_{111}'\varphi_{1\,rel}=\) | \(+c_{111}'\varphi_{1\,rel}^{2}=\) |
Izv. TPI, vol. 61, issue 2.